**Seoul National University**

Seoul National University System Health & Risk Management

**- 1 -**

## Chapter 14. Fatigue of Materials:

### Strain-Based Approach to Fatigue

Seoul National University System Health & Risk Management

### Jong Moon Ha

2015. 05. 06

**CHAPTER 14 Objectives**

### • **Explorer strain versus fatigue life curves and equations, including trends ** with material and adjustments for surface finish and size

### • **Extend strain-life curves to cases of nonzero mean stress and multi-axial ** **stress**

### • **Apply the strain-based method to make life estimates for engineering **

**components, especially members with geometric notches, including **

**cases of irregular variation of load with time**

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

** Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.*

**6**

**6**

**Comparison of Methods***

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

**6**

**6**

**Comparison of Methods***

**Review: Crack Initiation and Propagation***

• **The fatigue life of a component is made up of initiation and propagation stages**

• The size of crack is usually unknown and often depends on the size of the component being analyzed (e.g. A: 0.1mm-crack, B:0.15mm-crack)

• **Stress & Strain-based Approach: To determine crack initiation life**

• **Fracture Mechanics: To estimate the crack propagation life**

** Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273.*

**Figure 3.1* Initiation and propagation **

portions of fatigue life **Figure 3.9* Extended service life **
of a cracked component

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

**6**

**6**

**Comparison of Methods***

**14.1 Introduction – Why strain-based approach?**

**Brief Review of Stress-based Approach**

• **Key Idea is to develop stress-life relationship **
*by employing elastic stress concentration *
*factors and empirical modifications thereof.*

• **Stress-based approach emphasizes nominal **
**stress, rather than local stresses and strains**

• **Stress-based approach does not account for **
**plastic strain.**

**Figure 9.5**

**Stress-based Approach vs Strain-based Approach**

**High cycle fatigue** **Low cycle fatigue**

Elastic range of the material

(Does not account for plastic strain) Account for plastic strain More than 1000 cycles Less than 1000 cycles Stress-Life (Stress controlled) Strain-Life (Strain controlled)

Source: http://www.public.iastate.edu/~gkstarns/

**14.1 Introduction**

• **Employment of cyclic stress-strain curve is a unique feature of the strain-based ****approach, as is the use of a strain versus life curve, instead of a nominal stress ***versus life (S-N) curve*

• **Strain-based method gives improved estimates for intermediate and **
**especially short fatigue lives.**

• **Strain-based approach employs the local mean stress at the notch, rather than the **
mean nominal stress.

• **Certain concepts employed will be related to those introduced in Chapter 9 and 10. **

**Also, we will draw upon the information in Chapter 12 and 13 on plastic deformation**

Be familiar with the contents in Chapter 9, 10, 12 and 13 before studying Chapter 14.

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

**6**

**6**

**Comparison of Methods***

**14.2 Strain Versus Life Curves**

• **A strain versus life curve is a plot of strain amplitude versus cycles to failure **
(See Fig. 14.3)

**Figure 14.3 Elastic, plastic, and total strain versus life curves. (Adapted from [Landgraf 70]; copyright © ASTM; **

reprinted with permission.)

**14.2.1 Strain-Life Tests and Equations**

• Of particular relevance is the cyclic stress-strain curve (Also can be found in Eq. 12.54)

𝜀𝜀_{𝑎𝑎} = σ_{𝑎𝑎}
𝐸𝐸 +

σ_{𝑎𝑎}
𝐻𝐻^{′}

1/𝑛𝑛^{′}

(14.1)

𝜀𝜀_{𝑎𝑎} = 𝜀𝜀_{𝑒𝑒𝑎𝑎}+𝜀𝜀_{𝑝𝑝𝑎𝑎} (14.2)

Measure of the half-width of the stress-strain hysteresis loop

• Note that the strain amplitude can be divided into elastic and plastic parts as

𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_{𝑎𝑎}
𝐸𝐸

• Strain-life curves are derived from fatigue tests under completely reversed cyclic loading between constant strain limits (See Fig. 14.2)

**Figure 14.2**

Measurement:

𝜎𝜎_{𝑎𝑎}, 𝜀𝜀_{𝑎𝑎}, 𝜀𝜀_{𝑝𝑝𝑎𝑎}

**Figure 12.17**

𝜎𝜎_{𝑎𝑎} = ∆𝜎𝜎/2
𝜀𝜀_{𝑎𝑎} = ∆𝜀𝜀/2

**14.2.1 Strain-Life Tests and Equations**

• For each test, three points are plotted (Fig. 14.4)

• If data from several tests are plotted, the elastic
**strains** often give a straight line of shallow slope
on a log-log plot, and the plastic strains give a
straight line of steeper slope.

• Equation can then be fitted to these lines
𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_{𝑎𝑎}

𝐸𝐸 =
σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏

𝜀𝜀_{𝑝𝑝𝑎𝑎} = 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐}

**Figure 14.4 Strain versus life curves for RQC-**
100 steel. For each of several tests, elastic,
plastic, and total strain data points are plotted
versus life, and fitted lines are also shown.

(From the author’s data on the ASTM Committee E9 material.)

(14.3) – (a) (14.3) – (b)

• Four constants(σ_{𝑓𝑓}^{′}, 𝑏𝑏, 𝜀𝜀_{𝑓𝑓}^{′}, 𝑐𝑐) are considered to be
material properties

• Coffin-Manson Relationship
𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.4)

**①**

**③**

**②**

**Plastic**
**Elastic**

(𝜎𝜎_{𝑎𝑎}/𝐸𝐸)

**Remind - 14.2 Strain Versus Life Curves**

**Figure 14.3 Elastic, plastic, and total strain versus life curves. (Adapted from [Landgraf 70]; copyright © ASTM; **

reprinted with permission.)

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.4)

Four constants (σ_{𝑓𝑓}^{′}, 𝑏𝑏, 𝜀𝜀_{𝑓𝑓}^{′}, 𝑐𝑐)

• Coffin-Manson Relationship

**Plastic**
**Elastic**

**14.2.3 Trends for Engineering Metals**

• **At long lives: the curve approaches the elastic strain line**

• **At short lives: the curve approaches the plastic strain line**

• **Near the crossing point: the two types of strain are of similar magnitude**

*Transition Fatigue Life, 𝑁𝑁*_{𝑡𝑡}

• 𝑁𝑁_{𝑡𝑡} **is the most logical point for separating**
**low-cycle fatigue and high-cycle fatigue**

• **Special analysis of plasticity effects by the strain-based approach may be needed if **
**lives around or less than 𝑵𝑵**_{𝒕𝒕} are of interest

𝑁𝑁_{𝑡𝑡} = 1
2

𝜎𝜎_{𝑓𝑓}^{′}
𝜀𝜀_{𝑓𝑓}^{′}𝐸𝐸

1/(𝑐𝑐−𝑏𝑏)

Let 𝜀𝜀_{𝑒𝑒𝑎𝑎} = 𝜀𝜀_{𝑝𝑝𝑎𝑎}

(14.6)

Strain-based Approach

**14.2.3 Trends for Engineering Metals**

• The strain-life equation requires four empirical constants (σ_{𝑓𝑓}^{′}, 𝑏𝑏, 𝜀𝜀_{𝑓𝑓}^{′}, 𝑐𝑐)

• **Several points must be considered in attempting to obtain these constants from **
fatigue data*

** Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.*

**1) Generalization**

Not all materials may be represented by the four-parameter strain-life equation.

(Examples: some high strength aluminum alloys and titanium alloys)
**2) Data Size**

**The four fatigue constants may represent a curve fit to a limited number of data **
**points. They may be changed if more data points are included in the curve fit.**

**3) Range of data**

**The fatigue constants are determined from a set of data points over a given range.**

**Gross error may occur when extrapolating fatigue life estimates outside this range.**

**4) Physical phenomenon**

The use of this equation is strictly a matter of mathematical convenience
**and is not based on a physical phenomenon**

Nevertheless, the following approximate methods may be useful (cont.).

**14.2.3 Trends for Engineering Metals**

** Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p3 64.*

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐}

**Fatigue Strength Coefficient, 𝝈𝝈**_{𝒇𝒇}^{′} **Fatigue Ductility Coefficient, 𝜺𝜺**_{𝒇𝒇}^{′}
Approximation* σ^{𝑓𝑓}^{′} ≈ 𝜎𝜎_{𝑓𝑓} (corrected for necking)

For steels (<500 BHN): σ_{𝑓𝑓}^{′} ≈ 𝑆𝑆_{𝑢𝑢} + 50𝑘𝑘𝑘𝑘𝑘𝑘 𝜀𝜀_{𝑓𝑓}^{′} ≈ 𝜀𝜀_{𝑓𝑓}, where 𝜀𝜀_{𝑓𝑓} = ln _{1−RA}^{1}

RA is the reduction in area

A ductile Metal Low High

A brittle metal High Low

• All pass near the strain 𝜀𝜀_{𝑎𝑎} = 0.01 for a life of 𝑁𝑁_{𝑓𝑓} = 1000 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑘𝑘

**(Brittle)**
**High plasticity**

**Figure 14.6 Trends in strain–life **
curves for strong, tough, and ductile
metals. (Adapted from [Landgraf
70]; copyright © ASTM; reprinted
with permission.)

**14.2.3 Trends for Engineering Metals**

• * Fatigue Strength Exponent, b*
Average: -0.085

For soft metals: 𝑏𝑏 ≈ −0.12

For highly hardened metals: 𝑏𝑏 ≈ −0.05

For steels with ultimate tensile strengths below about 𝜎𝜎_{𝑢𝑢} = 1400MPa

(Fatigue limit occurs near 𝑁𝑁_{𝑓𝑓} = 10^{6} cycles at a stress amplitude around 𝜎𝜎_{𝑎𝑎} = 𝜎𝜎_{𝑢𝑢}/2)

In other case, where the fatigue limit (or long-life fatigue strength) at cycles is given
by 𝜎𝜎_{𝑎𝑎} = 𝑚𝑚_{𝑒𝑒}𝜎𝜎_{𝑢𝑢}, the estimate becomes

𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_{𝑎𝑎}
𝐸𝐸 =

σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 (14.3) – (a)

𝑏𝑏 = − 1

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_{𝑓𝑓} 𝑐𝑐𝑙𝑙𝑙𝑙 2σ_{𝑓𝑓}^{′}

σ_{𝑢𝑢} = − 1

6.3 𝑐𝑐𝑙𝑙𝑙𝑙

2�σ_{𝑓𝑓}

σ_{𝑢𝑢} (14.10)

𝑏𝑏 = − 1

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_{𝑓𝑓} 𝑐𝑐𝑙𝑙𝑙𝑙 �σ_{𝑓𝑓}

𝑚𝑚_{𝑒𝑒}σ_{𝑢𝑢} (14.11)

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐}

**14.2.3 Trends for Engineering Metals**

• **Fatigue Ductility Exponent, 𝑐𝑐 ***

𝑐𝑐 is not well defined as the other parameters.

A rule of thumb approach must be followed rather than an empirical equation Coffin found 𝑐𝑐 to be about -0.5

Manson found 𝑐𝑐 to be about -0.6

Morrow found that 𝑐𝑐 varied between -0.5 and -0.7

Fairly ductile metals (where 𝜀𝜀_{𝑓𝑓} ≈ 1) have average values of 𝑐𝑐=-0.6
For strong metals (where 𝜀𝜀_{𝑓𝑓} ≈ 0.5) have average values of 𝑐𝑐=-0.5

** Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p3 64.*

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐}

**14.2.3 Trends for Engineering Metals**

• Hardness varies inversely with ductility

*so that N** _{t}* decreases as hardness is increased.

**Figure 14.8 Transition fatigue life versus hardness for a wide range of steels. (Adapted from [Landgraf 70]; **

copyright © ASTM; reprinted with permission.)

**14.2.4 Factors Affecting Strain-Life Curves**

• **Is a hostile chemical environment or elevated temperature is present,**
smaller numbers of cycles to failure are expected

• **Temperature**

: At temperature exceeding about half of the absolute melting temperature of a given material, nonlinear deformation due to time-dependent creep-relaxation behavior generally become significant

• **Residual Stress**

Residual stress are quickly removed by cycle-dependent relaxation if cyclic plastic strains are present

**These have only limited effect at lives around and below N***_{t}*.
Some additional comments on this topic are given in Chapter 15.

**14.2.4 Factors Affecting Strain-Life Curves** **: Surface Finish**

• **High-cycle fatigue (most of the life is spent initiating a crack)**

Surface finish is important

• **Fatigue with significant plastic strain (most of the life is spent growing crack size)**

Surface finish cannot have an effect

• **A reasonable method to include the effect of surface finish is to change only the **
**elastic slope b.**

• When the fatigue limit at 𝑁𝑁_{𝑐𝑐} cycles is given by 𝜎𝜎_{𝑎𝑎} = 𝑚𝑚_{𝑠𝑠}𝜎𝜎_{𝑒𝑒} ,
where 𝑚𝑚_{𝑠𝑠} is a surface effect factor, as in Chapter 10

𝑏𝑏_{𝑠𝑠} = − 1

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_{𝑓𝑓} 𝑐𝑐𝑙𝑙𝑙𝑙 �σ_{𝑓𝑓}

𝑚𝑚_{𝑠𝑠}σ_{𝑒𝑒} = 𝑏𝑏 + 𝑐𝑐𝑙𝑙𝑙𝑙 𝑚𝑚_{𝑠𝑠}

𝑐𝑐𝑙𝑙𝑙𝑙 2𝑁𝑁_{𝑒𝑒} (14.12)
𝜎𝜎_{𝑒𝑒} = 𝜎𝜎_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑒𝑒} ^{𝑏𝑏}

**14.2.4 Factors Affecting Strain-Life Curves** **: Size Effect**

• Size Effect (as discussed in Chapter 10) are also a concern in applying a strain- based approach to large-size members, but experimental data are limited

• A study suggested lowering the entire strain-life curve by this factors so that the
intercept constants 𝜎𝜎_{𝑓𝑓}^{′} and 𝜀𝜀_{𝑓𝑓}^{′} are replaced by reduced values and 𝜎𝜎_{𝑓𝑓𝑑𝑑}^{′} and 𝜀𝜀_{𝑓𝑓𝑑𝑑}^{′}

where 𝑚𝑚_{𝑑𝑑} for the shafts up to 200 mm in diameter (low-carbon and low-alloy
steels) was found to vary with shaft diameter as

(14.14)
𝜎𝜎_{𝑓𝑓𝑑𝑑}^{′} = 𝑚𝑚_{𝑑𝑑}𝜎𝜎_{𝑓𝑓}^{′}, 𝜀𝜀_{𝑓𝑓𝑑𝑑}^{′} = 𝑚𝑚_{𝑑𝑑}𝜀𝜀_{𝑓𝑓}^{′}

(14.13)
𝑚𝑚_{𝑑𝑑} = 𝑑𝑑

25.4𝑚𝑚𝑚𝑚

−0.093

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

**6**

**6**

**Comparison of Methods***

**14.3.1 Mean Stress Effects***

• **Cyclic fatigue properties of a material are obtained from completely reversed, **
**constant amplitude strain-controlled tests.**

However, components seldom experience this type of loading.

• **Mean stress effect can either increase the fatigue life with a nominally **
**compressive load or decrease it with a nominally tensile value (Fig. 2.21*)**

• **At high strain amplitude (0.5% to 1% or above), where plastic strains are **

**significant, mean stress relaxation occurs and the mean stress tends toward zero**

• Modifications to the strain-life equation have been made to account for mean stress effects by 1) Morrow, 2) Smith, Watson, and Topper (SWT), and 3) Walker.

** Bannantine, Julie. "Fundamentals of metal fatigue analysis." Prentice Hall, 1990, (1990): 273, p. 63.*

**Figure 2.21* Effect of mean stress on strain-life curve**

**14.3.3 Mean Stress Equation of Morrow***

• Morrow (1968) suggested to modify the elastic term in the strain-life equation

𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_{𝑎𝑎}
𝐸𝐸 =

σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏

(14.3) – (a) 𝜀𝜀_{𝑒𝑒𝑎𝑎} = σ_{𝑎𝑎}
𝐸𝐸 =

σ_{𝑓𝑓}^{′} − σ_{𝑚𝑚}

𝐸𝐸 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏}
𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐}

(14.4)

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′} − σ_{𝑚𝑚}

𝐸𝐸 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐}

(14.27)

**Figure 2.23* Morrow’s mean stress correction to **

the strain-life curve for a tensile mean **Figure 14.12 Family of strain–life curves given by the **
modified Morrow approach

**14.3.3 Mean Stress Equation of Morrow (modified)***

• **Morrow and Halford (1981) found that ratio of elastic to plastic strain in the equation **
**proposed by Morrow (1968) is dependent on mean stress, which is not true.**

• **They modified both the elastic and plastic terms of the strain-life equation to maintain **
**the independence of the elastic-plastic strain ratio from mean stress as:**

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′} − σ_{𝑚𝑚}

𝐸𝐸 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} + 𝜀𝜀_{𝑓𝑓}^{′} σ_{𝑓𝑓}^{′} − σ_{𝑚𝑚}
σ_{𝑓𝑓}^{′}

𝑐𝑐/𝑏𝑏

2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.24)

• Refer to the Appendix for more details

**Figure 2.25* Mean stress correction for **
independence of elastic/plastic strain ratio
from mean stress

**(SWT) Parameters, Walker Mean Stress Equation**

**14.3.5 Smith, Watson, and Topper (SWT) Parameters**
𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′ 2}

𝐸𝐸 2𝑁𝑁_{𝑓𝑓} ^{2𝑏𝑏} + σ_{𝑓𝑓}^{′}𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏+𝑐𝑐} (14.29)

where 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝜎𝜎_{𝑚𝑚} + 𝜎𝜎_{𝑎𝑎}

**14.3.5 Walker Mean Stress Equation**
𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸

1 − 𝑅𝑅 2

(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} + 𝜀𝜀_{𝑓𝑓}^{′} 1 − 𝑅𝑅
2

𝑐𝑐(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.33)

**14.3.5 Smith, Watson, and Topper (SWT) Parameters**

**Figure 14.13 Plot of the Smith, Watson, and Topper parameter versus life for the data of Fig. 14.10.**

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

• Fatigue under multi-axial loading where plastic deformations occur is currently an area of active research.

• Reasonable estimates are possible for relatively simple situations

**6**

**6**

**Comparison of Methods***

**14.4.1 Effective Strain Approach**

• Consider situations where all cyclic loadings have the same frequency and are either in-phase or 180° out-of-phase.

• Recall three-dimensional stress-strain relationships (Ch. 12)

̅𝜀𝜀 = �𝜎𝜎

𝐸𝐸 + ̅𝜀𝜀^{𝑝𝑝} (12.23)

̅𝜀𝜀𝑝𝑝 = 2/3 × 𝜀𝜀_{𝑝𝑝1} − 𝜀𝜀_{𝑝𝑝2} ^{2} + 𝜀𝜀_{𝑝𝑝2} − 𝜀𝜀_{𝑝𝑝3} ^{2} + 𝜀𝜀_{𝑝𝑝3} − 𝜀𝜀_{𝑝𝑝1} ^{2} (12.22)

• Then, we can define an effective strain amplitude

�𝜎𝜎 = 1/ 2 × 𝜎𝜎_{1} − 𝜎𝜎_{2} ^{2} + 𝜎𝜎_{2} − 𝜎𝜎_{3} ^{2} + 𝜎𝜎_{3} − 𝜎𝜎_{1} ^{2} (12.21)

̅𝜀𝜀𝑎𝑎 = �𝜎𝜎_{𝑎𝑎}

𝐸𝐸 + ̅𝜀𝜀^{𝑝𝑝𝑎𝑎} (14.34)

**14.4.1 Effective Strain Approach**

̅𝜀𝜀𝑎𝑎 = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.35)

�𝜎𝜎_{𝑎𝑎} = σ_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏}, _{𝑝𝑝𝑎𝑎}̅𝜀𝜀 = 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} (14.36)

• Consider the special case of plane stress (Refer to Ch. 12.3.4 Application to Plane Stress)
𝜎𝜎_{2𝑎𝑎} = 𝜆𝜆𝜎𝜎_{1𝑎𝑎}, 𝜎𝜎_{3𝑎𝑎} = 0, 𝜀𝜀_{1𝑎𝑎} = 𝜀𝜀_{𝑒𝑒1𝑎𝑎} + 𝜀𝜀_{𝑝𝑝1𝑎𝑎}

• Combining Eqs. (12.19), (12.24) and (12.32) with Eqs. (14.35), (14.37)

(14.37)

𝜀𝜀_{1𝑎𝑎} =
σ_{𝑓𝑓}^{′}

𝐸𝐸 1 − 𝜈𝜈𝜆𝜆 2𝑁𝑁^{𝑓𝑓} ^{𝑏𝑏} + 𝜀𝜀_{𝑓𝑓}^{′} 1 − 0.5𝜆𝜆 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐}

1 − 𝜆𝜆 + 𝜆𝜆^{2} (14.38)

• For the special state of plane stress that is pure shear (𝜎𝜎_{2𝑎𝑎} = 𝜆𝜆𝜎𝜎_{1𝑎𝑎}, 𝜆𝜆 = −1)

• For uniaxial loading (𝜎𝜎_{𝟐𝟐} = 𝜎𝜎_{𝟑𝟑} = 0)

its value reduces to the uniaxial strain amplitude ( ̅𝜀𝜀_{𝑎𝑎} = 𝜀𝜀_{1𝒂𝒂})

𝛾𝛾_{𝑚𝑚𝑥𝑥𝑎𝑎} = σ_{𝑓𝑓}^{′}

3𝐺𝐺 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} + 3𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.39)

Shear strain amplitude Shear modulus Refer to Ch. 13.4 for more information

**14.4.2 Discussion of the Effective Strain Approach**

• Consider the hydrostatic stress

(14.40)
𝜎𝜎_{ℎ𝑎𝑎} = 𝜎𝜎_{1𝑎𝑎} + 𝜎𝜎_{2𝑎𝑎} + 𝜎𝜎_{3𝑎𝑎}

3

• Relative value of 𝜎𝜎_{ℎ𝑎𝑎} *my be expressed as a triaxiality factor for plane stress (𝜎𝜎*_{3𝑎𝑎} = 0)
𝑇𝑇 = 1 + 𝜆𝜆

1 − 𝜆𝜆 + 𝜆𝜆^{2} , (14.41)

(a) Pure planar shear (𝜆𝜆 = −1) 𝑇𝑇 = 0
(b) Uniaxial stress (𝜆𝜆 = 0) 𝑇𝑇 = 1
(c) Equal biaxial stress (𝜆𝜆 = 1) 𝑇𝑇 = 2
𝜎𝜎_{2𝑎𝑎} = 𝜆𝜆𝜎𝜎_{1𝑎𝑎},

• Marloff (1985) proposed to include this effect to strain-life equation as:

̅𝜀𝜀𝑎𝑎 = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 2^{1−𝑇𝑇}𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.42)

**14.4.3 Critical Plane Approaches**

• Critical plane approach is needed where the loading is non-proportional to a significant degree

• Stresses and strains normal to the crack plane may have a major effect on the behavior, accelerating the growth if they tend to open the crack.

1) Stresses and strains are determined for various orientations (planes) in the material.

2) Stresses and strains acting on the most severely loaded plane are used for analysis.

**14.4.3 Critical Plane Approaches**

• **Fatemi and Socie**
𝛾𝛾_{𝑎𝑎𝑐𝑐} 1 + 𝛼𝛼𝜎𝜎_{max𝑐𝑐}

σ_{𝑜𝑜}^{′} _{𝑎𝑎}̅𝜀𝜀 = 𝜏𝜏_{𝑓𝑓}^{′}

𝐺𝐺 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝛾𝛾_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.43)

where 𝛾𝛾_{𝑎𝑎𝑐𝑐} is the largest amplitude of shear strain for any plane,
𝜎𝜎_{max𝑐𝑐} is the peak tensile stress normal to the plane of 𝛾𝛾_{𝑎𝑎𝑐𝑐},
𝛼𝛼 is an empirical constant ranging from 0.6 to 1.0, and
σ_{𝑜𝑜}^{′} is the yield strength for the cyclic stress-strain curve.

𝜏𝜏_{𝑓𝑓}^{′}, 𝑏𝑏, 𝛾𝛾_{𝑓𝑓}^{′}, 𝑐𝑐 give the strain-life curve from completely reversed tests in pure shear.

(specifically torsion tests on thin-walled tubes)

**Figure 14.14 Crack under pure shear (a), where irregularities retard growth, compared with a situation (b) where a **
normal stress acts to open the crack, enhancing its growth. (Adapted from [Socie 87]; used with permission of ASME.)

**14.4.3 Critical Plane Approaches**

• **Fatemi and Socie**

(14.35)

• **Smith, Watson, and Topper (SWT) Parameters**
𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′ 2}

𝐸𝐸 2𝑁𝑁_{𝑓𝑓} ^{2𝑏𝑏} + σ_{𝑓𝑓}^{′}𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏+𝑐𝑐}

**The shortest life estimated from either Eqs. (14.43) or (14.35) is the final life estimate. **

𝛾𝛾_{𝑎𝑎𝑐𝑐} 1 + 𝛼𝛼𝜎𝜎_{max𝑐𝑐}

σ_{𝑜𝑜}^{′} _{𝑎𝑎}̅𝜀𝜀 = 𝜏𝜏_{𝑓𝑓}^{′}

𝐺𝐺 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝛾𝛾_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.43)

• A single multiaxial fatigue criterion that considers both the shear and normal stress cracking mode is that of Chu (1995)

(14.44)
2𝜏𝜏_{𝑚𝑚𝑎𝑎𝑚𝑚}𝛾𝛾_{𝑎𝑎} + 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_{𝑎𝑎} = 𝑓𝑓 𝑁𝑁_{𝑓𝑓}

where 𝑓𝑓 𝑁𝑁_{𝑓𝑓} can be obtained from uniaxial test data.

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

**6**

**6**

**Comparison of Methods***

**14.5.1 Constant Amplitude Loading**

• Assuming idealized behavior for the material

𝜀𝜀 = 𝑓𝑓 𝜎𝜎 , 𝜀𝜀_{𝑎𝑎} = 𝑓𝑓(𝜎𝜎_{𝑎𝑎}) (14.45)

(Monotonic) (Cyclic)

∆𝜀𝜀

2 = 𝑓𝑓

∆𝜎𝜎 2

(14.46)

**Figure 12.14 Stress–strain unloading and reloading behavior consistent with a spring and slider rheological model. **

The example curves plotted correspond to a Ramberg–Osgood stress–strain curve with constants as in Fig. 12.9.

**14.5.1 Constant Amplitude Loading**

• **Consider Neuber’s rule to analyze a notched member**

**Ch. 13 will be reviewed in the following two slides**

**Ch. 13 Stress-strain Analysis of ** **Plastically deforming Members**

**13.5.3 Estimates of Notch Stress and Strain for Local Yielding**

• Theoretical stress concentration factor, 𝐾𝐾_{𝑡𝑡}, is used to relate the nominal stress or
strain to the local values.*

• Upon yielding, the local values are no longer linearly related to the nominal values
by 𝐾𝐾_{𝑡𝑡}*

• Instead, the values are related in terms of stress and strain concentration factors as:

𝑘𝑘_{𝜎𝜎} = 𝜎𝜎 (Local stress)

S (Nominal stress) , 𝑘𝑘^{𝜀𝜀} = 𝜀𝜀 (Local strain)

e (Nominal strain) (13.56)

**Figure 14.13* Effect of yielding on 𝐾𝐾**_{𝜎𝜎} and 𝐾𝐾_{𝜖𝜖}

• After yielding, the actual local stress is less than
that predicted using 𝐾𝐾_{𝑡𝑡}

• After yielding, the actual local strain is greater
than that predicted using 𝐾𝐾_{𝑡𝑡}

• This is due to residual stresses at the notch root

**Ch. 13 Stress-strain Analysis of ** **Plastically deforming Members**

**13.5.3 Estimates of Notch Stress and Strain for Local Yielding**

• **Neuber’s rule states simply that the geometric mean of the stress and strain **
concentration factors remains equal to 𝑘𝑘_{𝑡𝑡} during plastic deformation

• If fully plastic yielding does not occur, e=S/E ^{applies.}

𝑘𝑘_{𝑡𝑡} = 𝑘𝑘_{𝜎𝜎}𝑘𝑘_{𝜀𝜀} (13.57)

𝜎𝜎𝜀𝜀 = 𝑘𝑘_{𝑡𝑡}𝑆𝑆 ^{2}

E ^{(13.58)}

**Figure 13.16 For a given notched **
member and stress–strain curve (a),
Neuber’s rule may be used to estimate
*local notch stresses and strains, σ and *
*ε, corresponding to a particular value *
*of nominal stress S. Stress and strain *
concentration factors vary as in (b).

**14.5.1 Constant Amplitude Loading**

• Consider Neuber’s rule to analyze a notched member
**Step 1**

**Step 2**

**Step 3**
**Step 4**

**Figure 14.15 Steps required **
in strain-based life prediction
for a notched member under
constant amplitude loading.

**14.5.1 Constant Amplitude Loading**

• **Step 1**

**14.5.1 Constant Amplitude Loading**

• **Step 2 & Step 3**

𝜀𝜀 = σ 𝐸𝐸 +

σ
𝐻𝐻^{′}

1/𝑛𝑛^{′}

𝜎𝜎𝜀𝜀 = 𝑘𝑘_{𝑡𝑡}𝑆𝑆 ^{2}
E

**14.5.1 Constant Amplitude Loading**

• **Step 4**

**14.5.2 Irregular Load Versus Time Histories**

• *The life to failure N** _{f}* corresponding to each hysteresis loop can be determined from its
combination of strain amplitude and mean stress.

• If the SWT parameter is used, 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} is simply the highest stress (𝜎𝜎_{𝐺𝐺}) for F-G-F’

• Having obtained the Nf value for each loop, we can apply the Palmgren Miner rule (Ch.

9), where each closed stress-strain hysteresis loop is considered to represent a cycle.

**Figure 14.16 Analysis of a notched member subjected to an irregular load versus time history. Notched **
member (a), made of 2024-T351 aluminum, is subjected to load history (b). The resulting local stress–strain
response at the notch is shown in (c).

**F-G-F’**

**Contents**

**0**

**0**

**Review: Crack Initiation and Propagation**

**1**

**1**

**Introduction**

**2**

**2**

**Strain Versus Life Curves**

**3**

**3**

**Mean Stress Effects**

**4**

**4**

**Multi-axial Stress Effects**

**5**

**5**

**Life Estimates For Structural Components**

**6**

**6**

**Comparison of Methods***

**6. Comparison of Methods***

**General Points for Comparison**

• **Are the methods to be used in the design cycle or to analyze an existing component?**

• What is the accuracy of the methods compared to input variables such as load history and material properties?

• What are the relative economics?

• What is the level of acceptance?

• Uses in design versus research.

**6. Comparison of Methods***

**Stress-based Approach** **Strain-based Approach**

**Economics** **Quick and Cheap** Expensive

**Available Data Size** **Big** Small

**History** **Long (100 years)** Short (30 years)

**Amount of Confidence** **High** Low

**Complexity** **Low** High

**Physical Insight** Bad **Good**

**Plastic Strain** Not considered **Considered**

**Residual Stress** Not considered **Considered**

**Where should we **

**use this approach?** • For constant amplitude loading
and long fatigue lives.

• When elastic strains are dominant (e.g. Transmission shaft, valve springs, and gears)

• For variable amplitude loading and short fatigue lives.

• When plastic strains are significant

• High temperature applications with fatigue-creep (e.g. Gas turbine engine)

**Appendix**

**14.3.2 Including Mean Stress Effects in Strain-Life Equation**

• *Let us define equivalent completely reversed stress amplitude 𝜎𝜎*_{𝑎𝑎𝑎𝑎} (as Eq. 9.22),
which is repeated here:

(14.15)
𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏}

• An additional equation is needed to calculate 𝜎𝜎_{𝑎𝑎𝑎𝑎} for the mean stress situation
(14.17)

𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝑓𝑓 𝜎𝜎_{𝑎𝑎}, 𝜎𝜎_{𝑚𝑚} = 𝜎𝜎_{𝑎𝑎} 𝑓𝑓 𝜎𝜎_{𝑎𝑎}, 𝜎𝜎_{𝑚𝑚}

𝜎𝜎_{𝑎𝑎} = 𝜎𝜎_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏}

• *Then, we can define zero-mean-stress-equivalent life, N**

𝜎𝜎_{𝑎𝑎} = 𝜎𝜎_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} 𝜎𝜎_{𝑎𝑎}
𝑓𝑓 𝜎𝜎_{𝑎𝑎}, 𝜎𝜎_{𝑚𝑚}

1/𝑏𝑏 𝑏𝑏

= 𝜎𝜎_{𝑓𝑓}^{′} 2𝑁𝑁^{∗ 𝑏𝑏}

where 𝑁𝑁^{∗} = 𝑁𝑁_{𝑓𝑓} _{𝑓𝑓 𝜎𝜎}^{𝜎𝜎}^{𝑎𝑎}

𝑎𝑎,𝜎𝜎_{𝑚𝑚}

1/𝑏𝑏

(14.18)

**14.3.3 Mean Stress Equation of Morrow***

• Equivalent completely reversed stress amplitude (Also can be found at Eq. (9.21)) (14.21)

𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_{𝑎𝑎}
1 − 𝜎𝜎𝜎𝜎^{𝑚𝑚}_{𝑓𝑓}^{′}

𝑁𝑁_{𝑚𝑚𝑚𝑚}^{∗} = 𝑁𝑁_{𝑓𝑓} 1 − 𝜎𝜎_{𝑚𝑚}
𝜎𝜎_{𝑓𝑓}^{′}

1/𝑏𝑏 (14.22)

𝑁𝑁_{𝑓𝑓} = 𝑁𝑁_{𝑚𝑚𝑚𝑚}^{∗} 1 − 𝜎𝜎_{𝑚𝑚}
𝜎𝜎_{𝑓𝑓}^{′}

−1/𝑏𝑏

(14.23)

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{∗ 𝑏𝑏} + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁^{∗ 𝑐𝑐}
𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}

𝐸𝐸 1 −
𝜎𝜎_{𝑚𝑚}

𝜎𝜎_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} + 𝜀𝜀_{𝑓𝑓}^{′} 1 − 𝜎𝜎_{𝑚𝑚}
𝜎𝜎_{𝑓𝑓}^{′}

𝑐𝑐/𝑏𝑏

2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.24)

**14.3.3 Mean Stress Equation of Morrow***

**Figure 14.11 Mean stress data of Fig. 14.10 plotted versus N**^{∗} according to the Morrow equation.

**14.3.5 Smith, Watson, and Topper (SWT) Parameters**

• This approach assumes that the life for any situation of mean stress depends on the product

where 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝜎𝜎_{𝑚𝑚} + 𝜎𝜎_{𝑎𝑎} and ℎ^{′′} 𝑁𝑁_{𝑓𝑓} indicates a function of fatigue life 𝑁𝑁_{𝑓𝑓}

• Life is expected to be the same as for completely reversed loading where this product has the same value.

• Let σ_{𝑎𝑎𝑎𝑎} and 𝜀𝜀_{𝑎𝑎𝑎𝑎} be the completely reversed stress and strain amplitude that result in
the same life 𝑁𝑁_{𝑓𝑓} as the 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}, 𝜀𝜀_{𝑎𝑎} combination.

• When 𝜎𝜎_{𝑚𝑚} = 0 and 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} = σ_{𝑎𝑎𝑎𝑎}, we find 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_{𝑎𝑎}=𝜎𝜎_{𝑎𝑎𝑎𝑎}𝜀𝜀_{𝑎𝑎𝑎𝑎}

• Then, using Eqs. 14.5 and 14.4, we can define

𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_{𝑎𝑎} = ℎ^{′′} 𝑁𝑁_{𝑓𝑓} (14.28)

𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} σ_{𝑓𝑓}^{′}

𝐸𝐸 2𝑁𝑁^{𝑓𝑓}

𝑏𝑏 + 𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.29)

𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′ 2}

𝐸𝐸 2𝑁𝑁_{𝑓𝑓} ^{2𝑏𝑏} + σ_{𝑓𝑓}^{′}𝜀𝜀_{𝑓𝑓}^{′} 2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏+𝑐𝑐} (14.30)

**14.3.5 Smith, Watson, and Topper (SWT) Parameters**

**Figure 14.13 Plot of the Smith, Watson, and Topper parameter versus life for the data of Fig. 14.10.**

**14.3.5 Walker Mean Stress Equation**

• Recall Eq. (9.19)

where 𝑅𝑅 = 𝜎𝜎_{𝑚𝑚𝑚𝑚𝑛𝑛}/𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}

𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}^{1−𝛾𝛾}𝜎𝜎_{𝑎𝑎}^{𝛾𝛾}, 𝜎𝜎_{𝑎𝑎𝑎𝑎} = 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} 1 − 𝑅𝑅 (14.31)

2

𝛾𝛾

𝑁𝑁_{𝑓𝑓} = 𝑁𝑁_{𝑚𝑚𝑚𝑚}^{∗} 1 − 𝜎𝜎_{𝑚𝑚}
𝜎𝜎_{𝑓𝑓}^{′}

−1/𝑏𝑏

(14.23)

𝑁𝑁_{𝑤𝑤}^{∗} = 𝑁𝑁_{𝑓𝑓} 𝜎𝜎_{𝑎𝑎}
𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚}

(1−𝛾𝛾)/𝑏𝑏

(14.32)
𝑁𝑁_{𝑤𝑤}^{∗} = 𝑁𝑁_{𝑓𝑓} 1 − 𝑅𝑅

2

(1−𝛾𝛾)/𝑏𝑏

𝜀𝜀_{𝑎𝑎} = σ_{𝑓𝑓}^{′}
𝐸𝐸

1 − 𝑅𝑅 2

(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_{𝑓𝑓} ^{𝑏𝑏} + 𝜀𝜀_{𝑓𝑓}^{′} 1 − 𝑅𝑅
2

𝑐𝑐(1−𝛾𝛾)/𝑏𝑏

2𝑁𝑁_{𝑓𝑓} ^{𝑐𝑐} (14.33)

**Backup**

**4.3 Strain-life Approach***

**4.3.1 Notch Root Stresses and Strains**

• Theoretical stress concentration factor, 𝐾𝐾_{𝑡𝑡}, is used to relate the nominal stress or
strain to the local values.

• Upon yielding, the local values are no longer linearly related to the nominal values
by 𝐾𝐾_{𝑡𝑡}

• Instead, the values are related in terms of stress and strain concentration factors as:

𝐾𝐾_{𝜎𝜎} = Local stress

Nominal stress 𝐾𝐾_{𝜖𝜖} = Local strain
Nominal strain

(4.11)* (4.12)*

**Figure 14.13* Effect of yielding on 𝐾𝐾**_{𝜎𝜎} and 𝐾𝐾_{𝜖𝜖}

• After yielding, the actual local stress is less than
that predicted using 𝐾𝐾_{𝑡𝑡}

• After yielding, the actual local strain is greater
than that predicted using 𝐾𝐾_{𝑡𝑡}

• This is due to residual stresses at the notch root

**14.1 Constant Amplitude Loading**

• Let strain be expressed as a function of 𝑆𝑆 that denotes load, moment, nominal stress, etc.

𝜀𝜀 = 𝑙𝑙 𝑆𝑆 (14.47)

𝜀𝜀_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝑙𝑙 𝑆𝑆_{𝑚𝑚𝑎𝑎𝑚𝑚} = 𝑓𝑓 𝜎𝜎_{𝑚𝑚𝑎𝑎𝑚𝑚} for tensile stress (14.48)

𝜀𝜀_{𝑎𝑎} = 𝑙𝑙 𝑆𝑆_{𝑎𝑎} = 𝑓𝑓 𝜎𝜎_{𝑎𝑎} ∆𝜀𝜀

2 = 𝑙𝑙

∆𝑆𝑆

2 = 𝑓𝑓

∆𝜎𝜎 2